A Simpler Proof of Kakutani's Conjecture on Random Subdivision and Its Generalizations

TL;DR

This paper presents a simplified and more general proof of Kakutani's conjecture on interval fragmentation, establishing limiting distributions and large-deviation bounds, with applications to random trees and heavy-tailed division schemes.

Contribution

It introduces a unified approach focusing on spacings rather than division points, providing stronger results and extending to arbitrary schemes and heavy-tailed distributions.

Findings

Limiting spacing distribution is independent of splitting scheme.

Partition points are uniformly distributed under certain conditions.

Results apply to random trees and heavy-tailed division densities.

Abstract

We shift the perspective on the interval fragmentation problem from division points to division spacings. This leads to a proof that is both simpler and stronger, establishing limiting distributions for partition points and spacings and, more importantly, including large-deviation error bounds. Moreover, the approach is general. We obtain the limiting spacing distribution for arbitrary division points, and claim it is independent of splitting scheme. It is shown that, under certain conditions, the limiting partition points are always uniformly distributed. Discrete case (heavy-tailed probability densities of division points) is investigated and new results are presented, such as the stage-wise progression of fragmentation process. The results obtained apply to random trees.